Eksamen august 2005

JMM

Solutions to the August 2005 exam

Problem 1

(1) u^{−1}(B) ⊂ (0, 1) ⊂ [0, 1] is open in [0, 1] (by continuity), (0, 1), and R (since (0, 1) is open in R). Any open and bounded subset of R is the union of at most countably many disjoint open intervals. (Open subsets of the locally path-connected space R have open pathcomponents. The open, path-connected, bounded subsets of R are open intervals. Each of these disjoint open intervals contains a rational number so there are at most countably many of them.)

(2) u^{−1}(0, 0) is a closed subset of the compact space [0, 1] so it is itself compact. It is covered by the open intervals (a_{i}, b_{i}). By compactness, it is covered by finitely many of them.

(3) By continuity of u, u(a_{i}) ∈ u([a_{i}, b_{i}]) = u(\overline{(a_{i}, b_{i})}) ⊂ \overline{u(a_{i}, b_{i})} ⊂ \overline{B}. As the intervals are disjoint, a_{i} ∉ u^{−1}(B), so u(a_{i}) ∉ B. Thus u(a_{i}) is in \overline{B} − B. Similarly for the other end-point, b_{i}.

Problem 2

The point O = (0, 0, 0) has the property that any neighborhood contains a connected neighborhood U such that U − {O} has three components. Only the four listed points have this property. Thus any self-homeomorphism of Δ must permute the four points. Any such permutation can be realized by an affine automorphism of R^{3}.

Problem 3

(1) The Heine-Borel theorem says that S^{1} and W are compact Hausdorff spaces since they are closed and bounded subspaces of R^{2}.

(2) The curve S^{1} − {0 × 1} has a parameterization of the form c_{2} : (0, 1) → R^{2} where c_{2} is a homeomorphism. Similarly, the curve W − [−1, 1] has a parameterization of the form c_{1} : (0, 1) → R^{2} where c_{1} is a homeomorphism. The maps c_{1} and c_{2} can be more or less explicit. The images of c_{1} and c_{2} are clearly dense in W and S^{1}, respectively.

(3) Define q: W → S^{1} by qc_{1}(t) = c_{2}(t) for all t ∈ R and q(W − c_{1}(R)) = S^{1} − c_{2}(R). The map q is continuous as we showed in (General Toplogy, Thm 21.3) or since the pre-image of the intersection of S^{1} with any open ball in R^{2} is open in W. (Consider separately the open subsets of S^{1} containing the point at infinity and those not containing the point at infinity.) The map q is a closed quotient map by the Closed map Lemma (General Toplogy, Lemma 18.8). Since q sends [−1, 1] to a point in S^{1} there is a factorization \overline{q}: W/[−1, 1] → S^{1} which is again a quotient map (General Topology, Thm 14.15) and, in fact, a homeomorphism as it is bijective (General Topology, Cor 14.9).

(4) Suppose that s_{2} exists. Then s_{2} is injective and, by the Closed map Lemma, an embedding of the compact space S^{1} into the Hausdorff space R. This contradicts the non-existence of such embeddings.

(5) Suppose that s_{1} exists. Then s_{1}([−1, 1]) ⊂ e^{−1}(1 × 0) = Z. By conectedness, s_{1}([−1, 1]) is a point. Thus s_{1} factors through W/[−1, 1] = S^{1} to give a map s_{2} as in the previous question. Since the map s_{2} does not exits, nor does s_{1}.